An optimal single-machine scheduling with linear deterioration rate and rate-modifying activities

This paper considers a single-machine scheduling with linear deterioration rate of processing speed and multiple rate-modifying activities simultaneously. A rate-modifying activity can change the processing rate of machine under consideration, which means after each rate-modifying activity the speed of the machine is fully recovered. The integration of these two concept is motivated by human operators and semi-automatic systems that experience performance degradation over time and require rate-modifying activities for recovery. The objective is to minimize the makespan. We need to decide the sequence of jobs and when to schedule the rate-modifying activities. An optimal schedule is proposed, which can solve the problem in O(nlogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n\log n)$$\end{document} time where n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} is the number of jobs.